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ACT Math : How to find the length of an arc

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Example Questions

Example Question #1 : How To Find The Length Of An Arc

Circle

In the circle above, the angle A in radians is $\frac{\pi}{4}$

What is the length of arc A?

Possible Answers:

$\pi r$

$\frac{\pi r}{8}$

$\frac{\pi r}{16}$

$\frac{\pi r}{4}$

$\frac{\pi r}{2}$

Correct answer:

$\frac{\pi r}{4}$

Explanation:

Circumference of a Circle = $2\pi r$

Arc Length

$=\frac{Angle A}{2\pi}\ast2\pi r$

$=\frac{\frac{\pi}{4}}{2\pi}\ast2\pi r$

$=\frac{\pi}{(4)2\pi}\ast2\pi r$

$=\frac{1}{8}\ast2\pi r=\frac{\pi r}{4}$

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Example Question #1 : How To Find The Length Of An Arc

_arc1

The figure above is a circle with center at and a radius of $15\:cm$ . This figure is not drawn to scale.

What is the length of the arc ABC in the figure above?

Possible Answers:

$30\pi\:cm$

$\frac{15\pi}{8}\:cm$

$75\pi\:cm$

$15\pi\:cm$

$\frac{25\pi}{4}\:cm$

Correct answer:

$\frac{25\pi}{4}\:cm$

Explanation:

Recall that the length of an arc is merely a percentage of the circumference. The circumference is found by the equation:

$C=2\pi r$

For our data, this is:

$C=2\pi * 15\:cm=30\pi\:cm$

Now the percentage for our arc is based on the angle $75^{\circ}$ and the total degrees in a circle, namely, $360^{\circ}$ .

So, the length of the arc is:

$\frac{75^{\circ}}{360^{\circ}} * 30\pi\:cm=\frac{75\pi}{12}\:cm=\frac{25\pi}{4}\:cm$

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Example Question #1 : How To Find The Length Of An Arc

If a circle has a circumference of , what is the measure of the arc contained by a degree angle located at the center of the circle?

Possible Answers:

$4\pi$

4.8

Correct answer:

Explanation:

A circle has a total of 360 degrees. If our angle is located at the center and is degrees, we can do $\frac{60}{360}=\frac{1}{6}$ to see that our angle makes up $\frac{1}{6}$ of the complete circle.

Therefore, our arc is going to be $\frac{1}{6}$ of our total circumference. $24\cdot \frac{1}{6}=\frac{24}{6}=4$

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Example Question #3 : How To Find The Length Of An Arc

What is the area of the sector of a circle with a central angle of 120 degrees and a radius of 5cm ? Simplify any fractions and leave your answer in terms of $\pi$ .

Possible Answers:

$\frac{3\pi}{25}cm^2$

$24\pi cm^2$

$3000\pi cm^2$

$25\pi \cm^2$

$\frac{25\pi}{3} cm^2$

Correct answer:

$\frac{25\pi}{3} cm^2$

Explanation:

The formula for the area of a sector of a circle is:

$\frac{central\: angle}{360}*\pi r^2$

The central angle given is 120 thus:

$\frac{120}{360}*\pi (5cm)^2= \frac{1}{3}\pi 25cm^2 = \frac{25\pi}{3}cm^2$

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Example Question #51 : Circles

A water wheel turns a $120^{\circ}$ arc every minute. If the radius of the wheel is , how far in meters does the wheel turn along its edge each minute?

Possible Answers:

$\frac{5\pi}{3} m$

$8\pi m$

$10\pi m$

$\frac{\pi}{3} m$

$4\pi m$

Correct answer:

$4\pi m$

Explanation:

If the radius is , then the circumference of the wheel is:

$C = 2r\pi = 12\pi$

If the wheel turns $120^{\circ}$ each minute, then it turns $\frac{120^{\circ}}{360^{\circ}} = \frac{1}{3}$ of the circumference each minute.

$d = 12\pi \cdot \frac{1}{3} = 4\pi$

Thus, the wheel turns $4\pi m$ each minute.

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Example Question #4 : How To Find The Length Of An Arc

What is the length of the arc ABC ?

arc2

The total area of the circle is $64\pi \, in^{2}$ and the area of the shaded region is $12\pi \,in^{2}$ .

Possible Answers:

$8\:in$

$\frac{12\pi}{5}\:in$

$\frac{3\pi}{16}\:in$

$16\pi\:in$

$3\pi\:in$

Correct answer:

$3\pi\:in$

Explanation:

If the area of the circle is $64\pi\:in^2$ , the radius can be found using the formula for the area of a circle:
$A=\pi r^2$

For our data, this is:

$64\pi=\pi r^2$

r^2=64

Therefore, $r=8\:in$

Now, the circumference of the circle is defined as:

$C=2\pi r$

For our data, this is:

$C=2*8*\pi=16\pi\:in$

Now, we know that a sector is a percentage of the total area. This percentage is easily calculated:

$\frac{12\pi\:in^2}{64\pi\:in^2}=\frac{3}{16}$

So, the length of the arc will merely be the same percentage, but now applied to the circumference:

$\frac{3}{16} * 16\pi\:in = 3\pi\:in$

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ACT Math : How to find the length of an arc

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #1 : How To Find The Length Of An Arc

Example Question #1 : How To Find The Length Of An Arc

Example Question #1 : How To Find The Length Of An Arc

Example Question #3 : How To Find The Length Of An Arc

Example Question #51 : Circles

Example Question #4 : How To Find The Length Of An Arc

All ACT Math Resources

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